Problem Set Workbook
Access the downloadable workbook for 2011 AIME II problems here.
Discussion Forum
Engage in discussion about the 2011 AIME II math contest by visiting Random Math AIME II 2011 Forum
Individual Problems and Solutions
For problems and detailed solutions to each of the 2011 AIME II problems, please refer below:
Problem 1: Gary purchased a large beverage, but drank only of this beverage, where and are relatively prime positive integers. If Gary had purchased only half as much and drunk twice as much, he would have wasted only as much beverage. Find .
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Problem 2: On square point lies on side and point lies on side , so that . Find the area of square .
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Problem 3: The degree measures of the angles of a convex -sided polygon form an increasing arithmetic
sequence with integer values. Find the degree measure of the smallest angle.
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Problem 4: In triangle . The angle bisector of intersects at point , and point is the midpoint of . Let be the point of intersection of and line . The ratio of to can be expressed in the form , where and are relatively prime positive integers. Find .
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Problem 5: The sum of the first terms of a geometric series is . The sum of the first terms of the same series is . Find the sum of the first terms of the series.
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Problem 6: Define an ordered quadruple of integers to be interesting if and . How many interesting ordered quadruples are there?
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Problem 7: Ed has five identical green marbles and a large supply of identical red marbles. He arranges the green marbles and some of the red marbles in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves equals the number of marbles whose right hand neighbor is the other color. An example of such an arrangement is . Let be the maximum number of red marbles for which Ed can make such an arrangement, and let be the number of ways in which Ed can arrange the marbles to satisfy the requirement. Find the remainder when is divided by .
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Problem 8: Let be the 12 zeros of the polynomial . For each , let be one of or . Then the maximum possible value of the real part of can be written as , where and are positive integers. Find .
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Problem 9: Let be nonnegative real numbers such that , and . Let and be positive relatively prime integers such that is the maximum possible value of . Find .
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Problem 10: A circle with center has radius . Chord of length and chord of length intersect at point . The distance between the midpoints of the two chords is . The quantity can be represented as , where and are relatively prime positive integers. Find the remainder when is divided by .
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Problem 11: Let be the matrix with entries as follows: for , ; for ; all other entries in are zero. Let be the determinant of matrix . Then can be represented as , where and are relatively prime positive integers. Find .
Note: The determinant of the matrix is , and the determinant of the matrix
for , the determinant of an matrix with first row or first column is equal to , where is the determinant of the matrix formed by eliminating the row and column containing .
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Problem 12: Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be , where and are relatively prime positive integers. Find .
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Problem 13: Point lies on the diagonal of square with . Let and be the circumcenters of triangles and , respectively. Given that and , then , where and are positive integers. Find .
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Problem 14: There are permutations of such that for divides for all integers with . Find the remainder when is divided by .
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Problem 15: Let . A real number is chosen at random from the interval . The probability that is equal to , where , and are positive integers, and none of , or is divisible by the square of a prime. Find .
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The problems on this page are the property of the MAA's American Mathematics Competitions